GM Lesson 054 Transposing Simple Non-linear Formulae

Learning Intentions

By the end of this lesson, students will be able to:

• Transpose simple non-linear formulae where necessary.
• Identify inverse operations needed to isolate a pronumeral.
• Use transposed formulae to calculate unknown quantities.

Prerequisites

Students should already be able to:

• Substitute values into formulae.
• Solve simple linear equations.
• Use inverse operations to isolate an unknown.
• Evaluate expressions involving squares, square roots and powers.
• Recognise common measurement formulae such as:

Key Idea Summary

A formula is transposed when it is rearranged to make a different pronumeral the subject.

For simple non-linear formulae, the unknown may be squared or cubed. The inverse operation must be used to undo this.

Operation on unknown	Inverse operation
	subtract
	add
	divide by
	multiply by
	square root
	cube root

When taking a square root in a measurement context, use the positive value because lengths, radii and heights cannot be negative.

For example, from :

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

To transpose a simple non-linear formula:

1. Identify the pronumeral that must become the subject.
2. Decide what operations are being applied to that pronumeral.
3. Undo those operations in reverse order.
4. Use square roots or cube roots where needed.
5. Substitute values only after the formula has been rearranged, unless the problem is easier by substituting first.
6. Check that the answer is reasonable in context.

Worked Example 1: Transposing the Area of a Circle Formula

The area of a circle is given by:

Transpose the formula to make the subject.

Therefore:

Worked Example 2: Finding the Radius of a Circle

A circular garden has area . Find the radius, correct to one decimal place.

Use:

Substitute .

The radius is approximately .

Worked Example 3: Transposing the Volume of a Cylinder Formula

The volume of a cylinder is given by:

Transpose the formula to make the subject.

Therefore:

Worked Example 4: Finding the Radius of a Cylinder

A cylinder has volume and height . Find the radius, correct to one decimal place.

Use:

Substitute and .

The radius is approximately .

Worked Example 5: Transposing Pythagoras’ Theorem

Pythagoras’ theorem is:

Transpose the formula to make the subject.

Therefore:

Worked Example 6: Finding a Shorter Side in a Right-Angled Triangle

A right-angled triangle has hypotenuse and one shorter side . Find the other shorter side.

Use:

Substitute and .

The missing side is .

Understanding Checks

Understanding Check 1

Which inverse operation is needed to isolate in the formula ?

Expected response: Divide by , then take the square root.

Understanding Check 2

A student rearranges as:

Explain the error.

Expected response: The division by must occur before taking the square root. The correct rearrangement is:

Understanding Check 3

Transpose to make the subject.

Expected working:

Understanding Check 4

Why do we usually reject the negative square root when solving for a radius or length?

Expected response: A physical length, radius or height cannot be negative.

Exercises

Simple Familiar Exercises

Exercise 1

Transpose the formula to make the subject.

Exercise 2

Transpose the formula to make the subject.

Exercise 3

Transpose the formula to make the subject.

Exercise 4

Transpose the formula to make the subject.

Exercise 5

Transpose the formula to make the subject.

Exercise 6

Find the radius of a circle with area . Round to one decimal place.

Exercise 7

Find the height of a cylinder with volume and radius . Round to one decimal place.

Exercise 8

Find the radius of a cylinder with volume and height . Round to one decimal place.

Complex Familiar Exercises

Exercise 9

A circular water tank lid has area . Find its radius, correct to two decimal places.

Exercise 10

A cylindrical storage container has volume and height . Find its radius, correct to one decimal place.

Exercise 11

A right-angled triangle has hypotenuse and one shorter side . Find the other shorter side.

Exercise 12

A cylinder has radius and volume . Find its height, correct to one decimal place.

Exercise 13

The volume of a cone is given by:

Transpose the formula to make the subject.

Exercise 14

The volume of a cone is given by:

Transpose the formula to make the subject.

Exercise 15

A cone has volume and height . Find its radius, correct to one decimal place.

Homework Problems

Problem 1

Transpose to make the subject.

Problem 2

Find the radius of a circle with area . Round to one decimal place.

Problem 3

Transpose to make the subject.

Problem 4

A cylinder has volume and radius . Find its height, correct to one decimal place.

Problem 5

Transpose to make the subject.

Problem 6

A cylinder has volume and height . Find its radius, correct to one decimal place.

Problem 7

Transpose to make the subject.

Problem 8

A right-angled triangle has hypotenuse and one shorter side . Find the other shorter side.

Problem 9

The volume of a cone is:

Transpose the formula to make the subject.

Problem 10

A cone has volume and height . Find its radius, correct to one decimal place.

Next: GM Lesson 055 Tables of Values from Formulae